There is growing interest in mathematics learning progressions in early childhood education. Counting is a skill usually developed early in life. The application of the counting principles in early childhood typically entails counting objects. This poses challenges for learning about zero. Indeed, the word "zero" is seldom used in the…

This study examines the feasibility and potential advantages of using large language models, in particular GPT-4o, to perform partial credit grading of large numbers of student written responses to introductory level physics problems. Students were instructed to write down verbal explanations of their reasoning process when solving one conceptual…

Investigating students' thinking in classroom tasks, particularly in science and engineering, is essential for improving educational practices and advancing student learning. In this context, the notion of "Ways of Thinking" (WoT) has gained traction in STEM education, offering a framework to explore how students approach and solve…

The introduction of programming education in K-12 schools to promote computational thinking has attracted a great deal of attention from scholars and educators. Debugging code is a central skill for students, but is also a considerable challenge when learning to program. Learners at the K-12 level often lack confidence in programming debugging due…

With the growing popularity of computational thinking (CT) classes in K-12 schools, it is important to investigate public perceptions of these initiatives. Analyzing public discussions and opinions provides valuable insights that can inform future educational policies and reforms. In this paper, we collected questions and answers related to CT…

Computational Thinking (CT) has evolved as an essential competency for K-12 students, and programming practices are recognized as the key way to facilitate CT development. However, most studies of CT development in middle graders have focused on visual programming, lacking evidence to demonstrate the effectiveness of Python programming. Therefore,…

In this paper, we first focus on the sum of powers of the first n positive odd integers, T[subscript k](n)=1[superscript k]+3[superscript k]+5[superscript k]+...+(2n-1)[superscript k], and derive in an elementary way a polynomial formula for T[subscript k](n) in terms of a specific type of generalized Stirling numbers. Then we consider the sum of…

In the era of the "reproducibility crisis" and the "P-value controversy" new ways of presentation and interpretation of the results of a meta-analysis are desirable. One suggestion that has been made for single studies almost six decades ago and taken up now and then is the P-value function. For a given outcome, this function…

The purpose of this research note is to introduce a latent growth curve reconstruction approach based on the Tabu search algorithm. The approach algorithmically enables researchers to optimally determine at both the individual and the group levels the order of the polynomial needed to represent the latent growth curve model. The procedure is…

The acid-catalyzed dehydration of regioisomeric methylcyclohexanols is a classic organic chemistry experiment featured in a variety of laboratory textbooks and literature. The mechanistic details of this reaction have received an inconsistent and occasionally inaccurate treatment, wherein the reaction has been described as a mix of E1, E2-like,…

Gaussian-2-Blender is an open-source application programming interface (API) written in Python that allows for the conversion of Gaussian input files to 3D objects of different formats. This new tool was developed in response to the shortcomings of available programs to import Gaussian calculations into augmented reality (AR) or virtual reality…

Inconsistent filtering of force plate data persists in published studies. Previous evidence suggests that low pass filtering produces the largest error when calculating jump height, but used jump height from unfiltered force plate data as the criterion. This study investigated the effects of low pass filtering on jump height calculated from force…

We propose an algorithm that allows calculating the remainder and the quotient of division between polynomials over commutative coefficient rings, without polynomial long division. We use the previous results to determine the quadratic factors of polynomials over commutative coefficient rings and, in particular, to completely factorize in Z[x] any…

A reference interval provides a basis for physicians to determine whether a measurement is typical of a healthy individual. It can be interpreted as a prediction interval for a new individual from the overall population. However, a reference interval based on a single study may not be representative of the broader population. Meta-analysis can…

Any quadratic function has a line of symmetry going through its vertex; any cubic function has 1800 rotational symmetry around its point of inflection. However, polynomial functions of degree greater than three can be both symmetrical and asymmetrical (Goehle & Kobayashi, 2013). This work considers algebraic conversions of symmetrical quartic…